Boivin, André; Zhu, Changzhong On the completeness of the system \(\{z^{\tau_n}\}\) in \(L^{2}\). (English) Zbl 1014.30004 J. Approximation Theory 118, No. 1, 1-19 (2002). Summary: Given an unbounded domain \(\Omega\) located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers \(\{\tau_n\}\) \((n=1,2,\dots)\) satisfying \({n\over|\tau_n|} \to D\) as \(n\to\infty\) with \(0<D<\infty\), and \(|\arg (\tau_n)|<\alpha< {\pi\over 2}\), we obtain a completeness theorem for the system \(\{z^{\tau_n}\}\) \((n=1,2, \dots)\) in \(L^2_a[\Omega]\). The case with weight is also considered. Cited in 1 ReviewCited in 6 Documents MSC: 30B60 Completeness problems, closure of a system of functions of one complex variable 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis Keywords:mean square approximation; complex Müntz theorem; unbounded domain PDFBibTeX XMLCite \textit{A. Boivin} and \textit{C. Zhu}, J. Approx. Theory 118, No. 1, 1--19 (2002; Zbl 1014.30004) Full Text: DOI References: [1] Boivin, A.; Zhu, C., On the \(L^2\)-completeness of some systems of analytic functions, Complex Variables, 45, 273-295 (2001) · Zbl 1023.30008 [2] Carleman, T., Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Ark. Mat. Atron. Fys., 17, 1-30 (1923) · JFM 49.0708.03 [3] Cheney, E. W., Introduction to Approximation Theory (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0161.25202 [4] Dzhrbasian, M. M., Metric theorems on completeness and on the representation of analytic functions, Uspehi Mat. Nauk (N.S.), 5, 194-198 (1950) [5] Dzhrbasian, M. M., Some questions of the theory of weighted polynomial approximation in a complex domain, Mat. Sbornik (N.S.), 36, 353-440 (1955) [6] Gaier, D., Lectures on Complex Approximation (1985), Birkhäuser: Birkhäuser Boston Basel Stuttgart [7] Ja. Levin, B., Lectures on Entire Functions (1996), Amer. Math. Soc: Amer. Math. Soc Providence [8] Mergelyan, S. M., On the completeness of systems of analytic functions, Amer. Math. Soc. Transl. Ser. 2, 19, 109-166 (1962) · Zbl 0122.31601 [9] Rudin, W., Real and Complex Analysis (1987), McGraw-Hill: McGraw-Hill New York · Zbl 0925.00005 [10] X. Shen, On the closure {\(z^{τ_n\)}^{\(j\)}z\); X. Shen, On the closure {\(z^{τ_n\)}^{\(j\)}z\) [11] X. Shen, On approximation of functions in the complex plane by the system of functions {\(z^{τ_n\)}^{\(j\)}z\); X. Shen, On approximation of functions in the complex plane by the system of functions {\(z^{τ_n\)}^{\(j\)}z\) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.